\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  -1.0 & -7.00 x_{1} & +  2.00 x_{2} & -3.00 x_{3} & -8.00 x_{4} & -4.00 x_{5} & +  4.00 x_{6} & +  2.00 x_{7}\\
 x_{9}   &  0.0 & -9.00 x_{1} & -1.00 x_{2} & -2.00 x_{3} & -10.00 x_{4} &   & -10.00 x_{6} & +  2.00 x_{7}\\
 x_{10}   &  -1.0 & -3.00 x_{1} & +  4.00 x_{2} & -4.00 x_{3} & -5.00 x_{4} & +  4.00 x_{5} & +  7.00 x_{6} & -4.00 x_{7}\\
 x_{11}   &  -3.0 & -4.00 x_{1} & -9.00 x_{2} & -1.00 x_{3} & + 10.00 x_{4} & +  2.00 x_{5} & -7.00 x_{6} & +  9.00 x_{7}\\
 x_{12}   &  -1.0 & +  9.00 x_{1} & +  2.00 x_{2} &   & +  1.00 x_{4} & -6.00 x_{5} & +  7.00 x_{6} & +  1.00 x_{7}\\
 x_{13}   &  2.0 & -2.00 x_{1} & -4.00 x_{2} & +  3.00 x_{3} & +  9.00 x_{4} & -3.00 x_{5} &   & +  6.00 x_{7}\\
 x_{14}   &  1.0 & -7.00 x_{1} & +  4.00 x_{2} & -2.00 x_{3} & +  4.00 x_{4} & +  3.00 x_{5} & -4.00 x_{6} &   \\
 x_{15}   &  2.0 & +  4.00 x_{1} & -2.00 x_{2} & -5.00 x_{3} & +  5.00 x_{4} & +  4.00 x_{5} & -3.00 x_{6} & -2.00 x_{7}\\
 x_{16}   &  -1.0 & -5.00 x_{1} & -1.00 x_{2} & -6.00 x_{3} & + 10.00 x_{4} & +  5.00 x_{5} & +  8.00 x_{6} & + 10.00 x_{7}\\
\hline
z    &  0.0 & -2.00 x_{1} &   & +  5.00 x_{3} & -2.00 x_{4} & -2.00 x_{5} & -3.00 x_{6} & +  4.00 x_{7}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{7}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,16$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & +  7.00 y_{8} & +  9.00 y_{9} & +  3.00 y_{10} & +  4.00 y_{11} & -9.00 y_{12} & +  2.00 y_{13} & +  7.00 y_{14} & -4.00 y_{15} & +  5.00 y_{16}\\
 y_{2}   &  1.0 & -2.00 y_{8} & +  1.00 y_{9} & -4.00 y_{10} & +  9.00 y_{11} & -2.00 y_{12} & +  4.00 y_{13} & -4.00 y_{14} & +  2.00 y_{15} & +  1.00 y_{16}\\
 y_{3}   &  1.0 & +  3.00 y_{8} & +  2.00 y_{9} & +  4.00 y_{10} & +  1.00 y_{11} &   & -3.00 y_{13} & +  2.00 y_{14} & +  5.00 y_{15} & +  6.00 y_{16}\\
 y_{4}   &  1.0 & +  8.00 y_{8} & + 10.00 y_{9} & +  5.00 y_{10} & -10.00 y_{11} & -1.00 y_{12} & -9.00 y_{13} & -4.00 y_{14} & -5.00 y_{15} & -10.00 y_{16}\\
 y_{5}   &  1.0 & +  4.00 y_{8} &   & -4.00 y_{10} & -2.00 y_{11} & +  6.00 y_{12} & +  3.00 y_{13} & -3.00 y_{14} & -4.00 y_{15} & -5.00 y_{16}\\
 y_{6}   &  1.0 & -4.00 y_{8} & + 10.00 y_{9} & -7.00 y_{10} & +  7.00 y_{11} & -7.00 y_{12} &   & +  4.00 y_{14} & +  3.00 y_{15} & -8.00 y_{16}\\
 y_{7}   &  1.0 & -2.00 y_{8} & -2.00 y_{9} & +  4.00 y_{10} & -9.00 y_{11} & -1.00 y_{12} & -6.00 y_{13} &   & +  2.00 y_{15} & -10.00 y_{16}\\
\hline
z    &  -0 & +  1.00 y_{8} &   & +  1.00 y_{10} & +  3.00 y_{11} & +  1.00 y_{12} & -2.00 y_{13} & -1.00 y_{14} & -2.00 y_{15} & +  1.00 y_{16}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 8 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{15}   &  1.14528860385 & -0.32 y_{5} & -0.34 y_{7} & -0.44 y_{2} & +  0.16 y_{6} & +  1.76 y_{8} & +  1.11 y_{13} & -1.96 y_{14} & -0.20 y_{1} & -2.28 y_{16}\\
 y_{10}   &  1.43709916132 & -0.36 y_{5} & -0.48 y_{7} & -0.55 y_{2} & +  0.06 y_{6} & +  0.39 y_{8} & +  0.66 y_{13} & -2.75 y_{14} & -0.11 y_{1} & -4.97 y_{16}\\
 y_{3}   &  15.8096941293 & -3.76 y_{5} & -4.83 y_{7} & -5.86 y_{2} & +  1.41 y_{6} & + 14.72 y_{8} & +  6.34 y_{13} & -26.02 y_{14} & -1.77 y_{1} & -35.05 y_{16}\\
 y_{4}   &  9.9438825851 & -1.53 y_{5} & -3.01 y_{7} & -5.29 y_{2} & +  1.29 y_{6} & +  5.49 y_{8} & -0.53 y_{13} & -32.08 y_{14} & -0.40 y_{1} & -30.18 y_{16}\\
 y_{11}   &  0.497903305377 & -0.13 y_{5} & -0.22 y_{7} & -0.12 y_{2} & +  0.00 y_{6} & +  0.11 y_{8} & -0.36 y_{13} & -0.61 y_{14} & -0.04 y_{1} & -2.48 y_{16}\\
 y_{12}   &  1.72089294524 & -0.33 y_{5} & -0.61 y_{7} & -0.70 y_{2} & +  0.15 y_{6} & +  0.80 y_{8} & +  0.56 y_{13} & -2.84 y_{14} & -0.22 y_{1} & -4.83 y_{16}\\
 y_{9}   &  1.41847557967 & -0.30 y_{5} & -0.51 y_{7} & -0.66 y_{2} & +  0.19 y_{6} & +  0.63 y_{8} & +  0.77 y_{13} & -3.30 y_{14} & -0.14 y_{1} & -3.64 y_{16}\\
\hline
z    &  2.361124815 & -0.43 y_{5} & -1.06 y_{7} & -0.72 y_{2} & -0.10 y_{6} & -0.99 y_{8} & -4.09 y_{13} & -4.50 y_{14} & -0.04 y_{1} & -11.68 y_{16}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  0.43216576221 & +  0.32 x_{15} & +  0.36 x_{10} & +  3.76 x_{3} & +  1.53 x_{4} & +  0.13 x_{11} & +  0.33 x_{12} & +  0.30 x_{9}\\
 x_{7}   &  1.06068080908 & +  0.34 x_{15} & +  0.48 x_{10} & +  4.83 x_{3} & +  3.01 x_{4} & +  0.22 x_{11} & +  0.61 x_{12} & +  0.51 x_{9}\\
 x_{2}   &  0.724716329551 & +  0.44 x_{15} & +  0.55 x_{10} & +  5.86 x_{3} & +  5.29 x_{4} & +  0.12 x_{11} & +  0.70 x_{12} & +  0.66 x_{9}\\
 x_{6}   &  0.104588061174 & -0.16 x_{15} & -0.06 x_{10} & -1.41 x_{3} & -1.29 x_{4} & -0.00 x_{11} & -0.15 x_{12} & -0.19 x_{9}\\
 x_{8}   &  0.98766650222 & -1.76 x_{15} & -0.39 x_{10} & -14.72 x_{3} & -5.49 x_{4} & -0.11 x_{11} & -0.80 x_{12} & -0.63 x_{9}\\
 x_{13}   &  4.09077454366 & -1.11 x_{15} & -0.66 x_{10} & -6.34 x_{3} & +  0.53 x_{4} & +  0.36 x_{11} & -0.56 x_{12} & -0.77 x_{9}\\
 x_{14}   &  4.50419338925 & +  1.96 x_{15} & +  2.75 x_{10} & + 26.02 x_{3} & + 32.08 x_{4} & +  0.61 x_{11} & +  2.84 x_{12} & +  3.30 x_{9}\\
 x_{1}   &  0.0389738529847 & +  0.20 x_{15} & +  0.11 x_{10} & +  1.77 x_{3} & +  0.40 x_{4} & +  0.04 x_{11} & +  0.22 x_{12} & +  0.14 x_{9}\\
 x_{16}   &  11.6847557967 & +  2.28 x_{15} & +  4.97 x_{10} & + 35.05 x_{3} & + 30.18 x_{4} & +  2.48 x_{11} & +  4.83 x_{12} & +  3.64 x_{9}\\
\hline
z    &  -2.361124815 & -1.15 x_{15} & -1.44 x_{10} & -15.81 x_{3} & -9.94 x_{4} & -0.50 x_{11} & -1.72 x_{12} & -1.42 x_{9}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  0.43216576221 & +  0.32 x_{15} & +  0.36 x_{10} & +  3.76 x_{3} & +  1.53 x_{4} & +  0.13 x_{11} & +  0.33 x_{12} & +  0.30 x_{9}\\
 x_{7}   &  1.06068080908 & +  0.34 x_{15} & +  0.48 x_{10} & +  4.83 x_{3} & +  3.01 x_{4} & +  0.22 x_{11} & +  0.61 x_{12} & +  0.51 x_{9}\\
 x_{2}   &  0.724716329551 & +  0.44 x_{15} & +  0.55 x_{10} & +  5.86 x_{3} & +  5.29 x_{4} & +  0.12 x_{11} & +  0.70 x_{12} & +  0.66 x_{9}\\
 x_{6}   &  0.104588061174 & -0.16 x_{15} & -0.06 x_{10} & -1.41 x_{3} & -1.29 x_{4} & -0.00 x_{11} & -0.15 x_{12} & -0.19 x_{9}\\
 x_{8}   &  0.98766650222 & -1.76 x_{15} & -0.39 x_{10} & -14.72 x_{3} & -5.49 x_{4} & -0.11 x_{11} & -0.80 x_{12} & -0.63 x_{9}\\
 x_{13}   &  4.09077454366 & -1.11 x_{15} & -0.66 x_{10} & -6.34 x_{3} & +  0.53 x_{4} & +  0.36 x_{11} & -0.56 x_{12} & -0.77 x_{9}\\
 x_{14}   &  4.50419338925 & +  1.96 x_{15} & +  2.75 x_{10} & + 26.02 x_{3} & + 32.08 x_{4} & +  0.61 x_{11} & +  2.84 x_{12} & +  3.30 x_{9}\\
 x_{1}   &  0.0389738529847 & +  0.20 x_{15} & +  0.11 x_{10} & +  1.77 x_{3} & +  0.40 x_{4} & +  0.04 x_{11} & +  0.22 x_{12} & +  0.14 x_{9}\\
 x_{16}   &  11.6847557967 & +  2.28 x_{15} & +  4.97 x_{10} & + 35.05 x_{3} & + 30.18 x_{4} & +  2.48 x_{11} & +  4.83 x_{12} & +  3.64 x_{9}\\
\hline
z    &  2.9866798224 & +  0.78 x_{15} & +  1.14 x_{10} & + 17.48 x_{3} & + 10.05 x_{4} & +  0.54 x_{11} & +  1.79 x_{12} & +  1.74 x_{9}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  0.43216576221 & +  0.32 x_{15} & +  0.36 x_{10} & +  3.76 x_{3} & +  1.53 x_{4} & +  0.13 x_{11} & +  0.33 x_{12} & +  0.30 x_{9}\\
 x_{7}   &  1.06068080908 & +  0.34 x_{15} & +  0.48 x_{10} & +  4.83 x_{3} & +  3.01 x_{4} & +  0.22 x_{11} & +  0.61 x_{12} & +  0.51 x_{9}\\
 x_{2}   &  0.724716329551 & +  0.44 x_{15} & +  0.55 x_{10} & +  5.86 x_{3} & +  5.29 x_{4} & +  0.12 x_{11} & +  0.70 x_{12} & +  0.66 x_{9}\\
 x_{6}   &  0.104588061174 & -0.16 x_{15} & -0.06 x_{10} & -1.41 x_{3} & -1.29 x_{4} & -0.00 x_{11} & -0.15 x_{12} & -0.19 x_{9}\\
 x_{8}   &  0.98766650222 & -1.76 x_{15} & -0.39 x_{10} & -14.72 x_{3} & -5.49 x_{4} & -0.11 x_{11} & -0.80 x_{12} & -0.63 x_{9}\\
 x_{13}   &  4.09077454366 & -1.11 x_{15} & -0.66 x_{10} & -6.34 x_{3} & +  0.53 x_{4} & +  0.36 x_{11} & -0.56 x_{12} & -0.77 x_{9}\\
 x_{14}   &  4.50419338925 & +  1.96 x_{15} & +  2.75 x_{10} & + 26.02 x_{3} & + 32.08 x_{4} & +  0.61 x_{11} & +  2.84 x_{12} & +  3.30 x_{9}\\
 x_{1}   &  0.0389738529847 & +  0.20 x_{15} & +  0.11 x_{10} & +  1.77 x_{3} & +  0.40 x_{4} & +  0.04 x_{11} & +  0.22 x_{12} & +  0.14 x_{9}\\
 x_{16}   &  11.6847557967 & +  2.28 x_{15} & +  4.97 x_{10} & + 35.05 x_{3} & + 30.18 x_{4} & +  2.48 x_{11} & +  4.83 x_{12} & +  3.64 x_{9}\\
\hline
z    &  2.9866798224 & +  0.78 x_{15} & +  1.14 x_{10} & + 17.48 x_{3} & + 10.05 x_{4} & +  0.54 x_{11} & +  1.79 x_{12} & +  1.74 x_{9}\\
\end{array}\]


 $ x_{3} $ enters and $ x_{8} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  0.68464167253 & -0.13 x_{15} & +  0.26 x_{10} & -0.26 x_{8} & +  0.13 x_{4} & +  0.10 x_{11} & +  0.12 x_{12} & +  0.14 x_{9}\\
 x_{7}   &  1.38436023721 & -0.24 x_{15} & +  0.35 x_{10} & -0.33 x_{8} & +  1.21 x_{4} & +  0.18 x_{11} & +  0.35 x_{12} & +  0.30 x_{9}\\
 x_{2}   &  1.11806881764 & -0.26 x_{15} & +  0.40 x_{10} & -0.40 x_{8} & +  3.10 x_{4} & +  0.07 x_{11} & +  0.38 x_{12} & +  0.41 x_{9}\\
 x_{6}   &  0.00971621938553 & +  0.01 x_{15} & -0.02 x_{10} & +  0.10 x_{8} & -0.76 x_{4} & +  0.01 x_{11} & -0.07 x_{12} & -0.13 x_{9}\\
 x_{3}   &  0.067075417965 & -0.12 x_{15} & -0.03 x_{10} & -0.07 x_{8} & -0.37 x_{4} & -0.01 x_{11} & -0.05 x_{12} & -0.04 x_{9}\\
 x_{13}   &  3.66577880524 & -0.35 x_{15} & -0.49 x_{10} & +  0.43 x_{8} & +  2.89 x_{4} & +  0.41 x_{11} & -0.21 x_{12} & -0.50 x_{9}\\
 x_{14}   &  6.24925453144 & -1.15 x_{15} & +  2.06 x_{10} & -1.77 x_{8} & + 22.37 x_{4} & +  0.41 x_{11} & +  1.42 x_{12} & +  2.19 x_{9}\\
 x_{1}   &  0.157704291889 & -0.01 x_{15} & +  0.06 x_{10} & -0.12 x_{8} & -0.26 x_{4} & +  0.02 x_{11} & +  0.12 x_{12} & +  0.07 x_{9}\\
 x_{16}   &  14.035497705 & -1.91 x_{15} & +  4.04 x_{10} & -2.38 x_{8} & + 17.10 x_{4} & +  2.21 x_{11} & +  2.92 x_{12} & +  2.14 x_{9}\\
\hline
z    &  4.15897745167 & -1.31 x_{15} & +  0.68 x_{10} & -1.19 x_{8} & +  3.52 x_{4} & +  0.40 x_{11} & +  0.84 x_{12} & +  1.00 x_{9}\\
\end{array}\]


 $ x_{4} $ enters and $ x_{6} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  0.686260685644 & -0.13 x_{15} & +  0.26 x_{10} & -0.24 x_{8} & -0.17 x_{6} & +  0.10 x_{11} & +  0.11 x_{12} & +  0.11 x_{9}\\
 x_{7}   &  1.39984136776 & -0.22 x_{15} & +  0.31 x_{10} & -0.17 x_{8} & -1.59 x_{6} & +  0.19 x_{11} & +  0.24 x_{12} & +  0.09 x_{9}\\
 x_{2}   &  1.15766281837 & -0.21 x_{15} & +  0.31 x_{10} & -0.01 x_{8} & -4.08 x_{6} & +  0.11 x_{11} & +  0.11 x_{12} & -0.13 x_{9}\\
 x_{4}   &  0.0127787080286 & +  0.02 x_{15} & -0.03 x_{10} & +  0.13 x_{8} & -1.32 x_{6} & +  0.01 x_{11} & -0.09 x_{12} & -0.18 x_{9}\\
 x_{3}   &  0.0623072177668 & -0.13 x_{15} & -0.02 x_{10} & -0.12 x_{8} & +  0.49 x_{6} & -0.01 x_{11} & -0.02 x_{12} & +  0.02 x_{9}\\
 x_{13}   &  3.70274081255 & -0.31 x_{15} & -0.57 x_{10} & +  0.80 x_{8} & -3.80 x_{6} & +  0.44 x_{11} & -0.47 x_{12} & -1.01 x_{9}\\
 x_{14}   &  6.53511941482 & -0.81 x_{15} & +  1.42 x_{10} & +  1.06 x_{8} & -29.42 x_{6} & +  0.67 x_{11} & -0.57 x_{12} & -1.75 x_{9}\\
 x_{1}   &  0.154402044593 & -0.01 x_{15} & +  0.07 x_{10} & -0.15 x_{8} & +  0.34 x_{6} & +  0.02 x_{11} & +  0.15 x_{12} & +  0.11 x_{9}\\
 x_{16}   &  14.2539878382 & -1.65 x_{15} & +  3.55 x_{10} & -0.22 x_{8} & -22.49 x_{6} & +  2.41 x_{11} & +  1.39 x_{12} & -0.87 x_{9}\\
\hline
z    &  4.20401868335 & -1.26 x_{15} & +  0.58 x_{10} & -0.74 x_{8} & -4.64 x_{6} & +  0.45 x_{11} & +  0.52 x_{12} & +  0.37 x_{9}\\
\end{array}\]


 $ x_{9} $ enters and $ x_{4} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  0.694451387153 & -0.12 x_{15} & +  0.24 x_{10} & -0.16 x_{8} & -1.01 x_{6} & +  0.11 x_{11} & +  0.06 x_{12} & -0.64 x_{4}\\
 x_{7}   &  1.4063984004 & -0.21 x_{15} & +  0.30 x_{10} & -0.11 x_{8} & -2.27 x_{6} & +  0.20 x_{11} & +  0.20 x_{12} & -0.51 x_{4}\\
 x_{2}   &  1.14796300925 & -0.22 x_{15} & +  0.33 x_{10} & -0.10 x_{8} & -3.08 x_{6} & +  0.10 x_{11} & +  0.17 x_{12} & +  0.76 x_{4}\\
 x_{9}   &  0.0724818795301 & +  0.09 x_{15} & -0.16 x_{10} & +  0.72 x_{8} & -7.46 x_{6} & +  0.07 x_{11} & -0.51 x_{12} & -5.67 x_{4}\\
 x_{3}   &  0.063984003999 & -0.12 x_{15} & -0.02 x_{10} & -0.10 x_{8} & +  0.32 x_{6} & -0.01 x_{11} & -0.03 x_{12} & -0.13 x_{4}\\
 x_{13}   &  3.62971757061 & -0.39 x_{15} & -0.41 x_{10} & +  0.07 x_{8} & +  3.71 x_{6} & +  0.38 x_{11} & +  0.04 x_{12} & +  5.71 x_{4}\\
 x_{14}   &  6.40827293177 & -0.96 x_{15} & +  1.71 x_{10} & -0.19 x_{8} & -16.37 x_{6} & +  0.55 x_{11} & +  0.31 x_{12} & +  9.93 x_{4}\\
 x_{1}   &  0.162709322669 & -0.00 x_{15} & +  0.05 x_{10} & -0.07 x_{8} & -0.52 x_{6} & +  0.03 x_{11} & +  0.09 x_{12} & -0.65 x_{4}\\
 x_{16}   &  14.1908272932 & -1.72 x_{15} & +  3.70 x_{10} & -0.84 x_{8} & -15.99 x_{6} & +  2.36 x_{11} & +  1.83 x_{12} & +  4.94 x_{4}\\
\hline
z    &  4.23119220195 & -1.22 x_{15} & +  0.52 x_{10} & -0.47 x_{8} & -7.43 x_{6} & +  0.47 x_{11} & +  0.33 x_{12} & -2.13 x_{4}\\
\end{array}\]


 $ x_{10} $ enters and $ x_{9} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  0.800307219662 & +  0.01 x_{15} & -1.46 x_{9} & +  0.89 x_{8} & -11.90 x_{6} & +  0.21 x_{11} & -0.68 x_{12} & -8.92 x_{4}\\
 x_{7}   &  1.53917050691 & -0.05 x_{15} & -1.83 x_{9} & +  1.20 x_{8} & -15.93 x_{6} & +  0.32 x_{11} & -0.73 x_{12} & -10.90 x_{4}\\
 x_{2}   &  1.29493087558 & -0.05 x_{15} & -2.03 x_{9} & +  1.35 x_{8} & -18.20 x_{6} & +  0.24 x_{11} & -0.85 x_{12} & -10.74 x_{4}\\
 x_{10}   &  0.445468509985 & +  0.53 x_{15} & -6.15 x_{9} & +  4.40 x_{8} & -45.85 x_{6} & +  0.41 x_{11} & -3.11 x_{12} & -34.86 x_{4}\\
 x_{3}   &  0.0552995391705 & -0.13 x_{15} & +  0.12 x_{9} & -0.18 x_{8} & +  1.21 x_{6} & -0.02 x_{11} & +  0.03 x_{12} & +  0.55 x_{4}\\
 x_{13}   &  3.44854070661 & -0.61 x_{15} & +  2.50 x_{9} & -1.72 x_{8} & + 22.36 x_{6} & +  0.21 x_{11} & +  1.31 x_{12} & + 19.89 x_{4}\\
 x_{14}   &  7.16897081413 & -0.05 x_{15} & -10.50 x_{9} & +  7.33 x_{8} & -94.66 x_{6} & +  1.25 x_{11} & -5.00 x_{12} & -49.60 x_{4}\\
 x_{1}   &  0.185867895545 & +  0.02 x_{15} & -0.32 x_{9} & +  0.16 x_{8} & -2.90 x_{6} & +  0.05 x_{11} & -0.07 x_{12} & -2.46 x_{4}\\
 x_{16}   &  15.8371735791 & +  0.25 x_{15} & -22.71 x_{9} & + 15.43 x_{8} & -185.43 x_{6} & +  3.86 x_{11} & -9.66 x_{12} & -123.89 x_{4}\\
\hline
z    &  4.46082949309 & -0.95 x_{15} & -3.17 x_{9} & +  1.80 x_{8} & -31.07 x_{6} & +  0.68 x_{11} & -1.27 x_{12} & -20.10 x_{4}\\
\end{array}\]


 $ x_{8} $ enters and $ x_{3} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  1.06666666667 & -0.63 x_{15} & -0.88 x_{9} & -4.82 x_{3} & -6.07 x_{6} & +  0.12 x_{11} & -0.55 x_{12} & -6.28 x_{4}\\
 x_{7}   &  1.9 & -0.92 x_{15} & -1.05 x_{9} & -6.52 x_{3} & -8.03 x_{6} & +  0.20 x_{11} & -0.55 x_{12} & -7.33 x_{4}\\
 x_{2}   &  1.7 & -1.02 x_{15} & -1.15 x_{9} & -7.32 x_{3} & -9.33 x_{6} & +  0.10 x_{11} & -0.65 x_{12} & -6.73 x_{4}\\
 x_{10}   &  1.76666666667 & -2.66 x_{15} & -3.28 x_{9} & -23.89 x_{3} & -16.89 x_{6} & -0.03 x_{11} & -2.45 x_{12} & -21.76 x_{4}\\
 x_{8}   &  0.3 & -0.72 x_{15} & +  0.65 x_{9} & -5.42 x_{3} & +  6.58 x_{6} & -0.10 x_{11} & +  0.15 x_{12} & +  2.98 x_{4}\\
 x_{13}   &  2.93333333333 & +  0.63 x_{15} & +  1.38 x_{9} & +  9.32 x_{3} & + 11.07 x_{6} & +  0.38 x_{11} & +  1.05 x_{12} & + 14.78 x_{4}\\
 x_{14}   &  9.36666666667 & -5.36 x_{15} & -5.73 x_{9} & -39.74 x_{3} & -46.49 x_{6} & +  0.52 x_{11} & -3.90 x_{12} & -27.81 x_{4}\\
 x_{1}   &  0.233333333333 & -0.09 x_{15} & -0.22 x_{9} & -0.86 x_{3} & -1.86 x_{6} & +  0.03 x_{11} & -0.05 x_{12} & -1.99 x_{4}\\
 x_{16}   &  20.4666666667 & -10.93 x_{15} & -12.68 x_{9} & -83.72 x_{3} & -83.97 x_{6} & +  2.32 x_{11} & -7.35 x_{12} & -77.98 x_{4}\\
\hline
z    &  5.0 & -2.25 x_{15} & -2.00 x_{9} & -9.75 x_{3} & -19.25 x_{6} & +  0.50 x_{11} & -1.00 x_{12} & -14.75 x_{4}\\
\end{array}\]


 $ x_{11} $ enters and $ x_{8} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  1.41666666667 & -1.48 x_{15} & -0.13 x_{9} & -11.15 x_{3} & +  1.60 x_{6} & -1.17 x_{8} & -0.38 x_{12} & -2.81 x_{4}\\
 x_{7}   &  2.5 & -2.37 x_{15} & +  0.25 x_{9} & -17.37 x_{3} & +  5.12 x_{6} & -2.00 x_{8} & -0.25 x_{12} & -1.38 x_{4}\\
 x_{2}   &  2.0 & -1.75 x_{15} & -0.50 x_{9} & -12.75 x_{3} & -2.75 x_{6} & -1.00 x_{8} & -0.50 x_{12} & -3.75 x_{4}\\
 x_{10}   &  1.66666666667 & -2.42 x_{15} & -3.50 x_{9} & -22.08 x_{3} & -19.08 x_{6} & +  0.33 x_{8} & -2.50 x_{12} & -22.75 x_{4}\\
 x_{11}   &  3.0 & -7.25 x_{15} & +  6.50 x_{9} & -54.25 x_{3} & + 65.75 x_{6} & -10.00 x_{8} & +  1.50 x_{12} & + 29.75 x_{4}\\
 x_{13}   &  4.08333333333 & -2.15 x_{15} & +  3.87 x_{9} & -11.48 x_{3} & + 36.27 x_{6} & -3.83 x_{8} & +  1.63 x_{12} & + 26.19 x_{4}\\
 x_{14}   &  10.9166666667 & -9.10 x_{15} & -2.37 x_{9} & -67.77 x_{3} & -12.52 x_{6} & -5.17 x_{8} & -3.13 x_{12} & -12.44 x_{4}\\
 x_{1}   &  0.333333333333 & -0.33 x_{15} & +  0.00 x_{9} & -2.67 x_{3} & +  0.33 x_{6} & -0.33 x_{8} & -0.00 x_{12} & -1.00 x_{4}\\
 x_{16}   &  27.4166666667 & -27.73 x_{15} & +  2.38 x_{9} & -209.40 x_{3} & + 68.35 x_{6} & -23.17 x_{8} & -3.88 x_{12} & -9.06 x_{4}\\
\hline
z    &  6.5 & -5.87 x_{15} & +  1.25 x_{9} & -36.87 x_{3} & + 13.62 x_{6} & -5.00 x_{8} & -0.25 x_{12} & +  0.13 x_{4}\\
\end{array}\]


 $ x_{4} $ enters and $ x_{10} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  1.21062271062 & -1.18 x_{15} & +  0.31 x_{9} & -8.42 x_{3} & +  3.96 x_{6} & -1.21 x_{8} & -0.07 x_{12} & +  0.12 x_{10}\\
 x_{7}   &  2.39926739927 & -2.23 x_{15} & +  0.46 x_{9} & -16.04 x_{3} & +  6.28 x_{6} & -2.02 x_{8} & -0.10 x_{12} & +  0.06 x_{10}\\
 x_{2}   &  1.72527472527 & -1.35 x_{15} & +  0.08 x_{9} & -9.11 x_{3} & +  0.40 x_{6} & -1.05 x_{8} & -0.09 x_{12} & +  0.16 x_{10}\\
 x_{4}   &  0.0732600732601 & -0.11 x_{15} & -0.15 x_{9} & -0.97 x_{3} & -0.84 x_{6} & +  0.01 x_{8} & -0.11 x_{12} & -0.04 x_{10}\\
 x_{11}   &  5.17948717949 & -10.41 x_{15} & +  1.92 x_{9} & -83.13 x_{3} & + 40.79 x_{6} & -9.56 x_{8} & -1.77 x_{12} & -1.31 x_{10}\\
 x_{13}   &  6.00183150183 & -4.93 x_{15} & -0.15 x_{9} & -36.90 x_{3} & + 14.30 x_{6} & -3.45 x_{8} & -1.25 x_{12} & -1.15 x_{10}\\
 x_{14}   &  10.0054945055 & -7.78 x_{15} & -0.46 x_{9} & -55.70 x_{3} & -2.09 x_{6} & -5.35 x_{8} & -1.76 x_{12} & +  0.55 x_{10}\\
 x_{1}   &  0.260073260073 & -0.23 x_{15} & +  0.15 x_{9} & -1.70 x_{3} & +  1.17 x_{6} & -0.35 x_{8} & +  0.11 x_{12} & +  0.04 x_{10}\\
 x_{16}   &  26.7527472527 & -26.77 x_{15} & +  3.77 x_{9} & -200.60 x_{3} & + 75.96 x_{6} & -23.30 x_{8} & -2.88 x_{12} & +  0.40 x_{10}\\
\hline
z    &  6.50915750916 & -5.89 x_{15} & +  1.23 x_{9} & -37.00 x_{3} & + 13.52 x_{6} & -5.00 x_{8} & -0.26 x_{12} & -0.01 x_{10}\\
\end{array}\]


 $ x_{6} $ enters and $ x_{4} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  1.55676855895 & -1.68 x_{15} & -0.42 x_{9} & -13.00 x_{3} & -4.72 x_{4} & -1.14 x_{8} & -0.59 x_{12} & -0.08 x_{10}\\
 x_{7}   &  2.94759825328 & -3.02 x_{15} & -0.69 x_{9} & -23.31 x_{3} & -7.48 x_{4} & -1.91 x_{8} & -0.92 x_{12} & -0.27 x_{10}\\
 x_{2}   &  1.75982532751 & -1.40 x_{15} & +  0.00 x_{9} & -9.57 x_{3} & -0.47 x_{4} & -1.05 x_{8} & -0.14 x_{12} & +  0.14 x_{10}\\
 x_{6}   &  0.0873362445415 & -0.13 x_{15} & -0.18 x_{9} & -1.16 x_{3} & -1.19 x_{4} & +  0.02 x_{8} & -0.13 x_{12} & -0.05 x_{10}\\
 x_{11}   &  8.7423580786 & -15.58 x_{15} & -5.56 x_{9} & -130.34 x_{3} & -48.63 x_{4} & -8.85 x_{8} & -7.11 x_{12} & -3.45 x_{10}\\
 x_{13}   &  7.25109170306 & -6.74 x_{15} & -2.78 x_{9} & -53.45 x_{3} & -17.05 x_{4} & -3.20 x_{8} & -3.13 x_{12} & -1.90 x_{10}\\
 x_{14}   &  9.8231441048 & -7.52 x_{15} & -0.08 x_{9} & -53.28 x_{3} & +  2.49 x_{4} & -5.39 x_{8} & -1.48 x_{12} & +  0.66 x_{10}\\
 x_{1}   &  0.362445414847 & -0.38 x_{15} & -0.06 x_{9} & -3.05 x_{3} & -1.40 x_{4} & -0.33 x_{8} & -0.04 x_{12} & -0.02 x_{10}\\
 x_{16}   &  33.3864628821 & -36.39 x_{15} & -10.16 x_{9} & -288.50 x_{3} & -90.55 x_{4} & -21.97 x_{8} & -12.83 x_{12} & -3.58 x_{10}\\
\hline
z    &  7.68995633188 & -7.60 x_{15} & -1.25 x_{9} & -52.64 x_{3} & -16.12 x_{4} & -4.76 x_{8} & -2.03 x_{12} & -0.71 x_{10}\\
\end{array}\]
Final Dictionary
Solution:  7.68995633188
Num Pivots:  9
\end{document}
